Problem: $ -2.\overline{56} \div 1.\overline{44} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -256.5657...\\ x &= -2.5657...\end{align*} $ $\begin{align*} 99x &= -254 \\ x &= -\dfrac{254}{99}\end{align*} $ $\begin{align*} 100y &= 144.4444...\\ y &= 1.4444...\end{align*} $ $\begin{align*} 99y &= 143 \\ y &= \dfrac{143}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{254}{99} \div \dfrac{143}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{254}{99} \times \dfrac{99}{143} = {?} $ $ \phantom{-\dfrac{254}{99} \times \dfrac{143}{99}} = \dfrac{-254 \times 99}{99 \times 143} $ $ \phantom{-\dfrac{254}{99} \times \dfrac{143}{99}} = \dfrac{-254 \times \cancel{99}} {\cancel{99} \times 143} $ $ \phantom{-\dfrac{254}{99} \times \dfrac{143}{99}} = -\dfrac{254}{143} $